[EM] Relevance of Consistency

[Blake Cretney]

On Wed, 2002-11-06 at 19:05, Forest Simmons wrote: 

> Only in those cases where at least one of the subsets (of the CC
> hypothesis) has no CW is (the conclusion of) the Consistency Criterion
> violated.
> 
> But when one or more subsets is indecisive, i.e. does not give definite
> support to any opinion as to which candidate should win, we should not be
> surprised if the over all outcome disagrees with a common subset outcome
> whose commonality was just a statistical fluke.

I'm going to take advantage of your argument here, without necessarily
agreeing with it.  Let's imagine that groups can have opinions.  Your
argument, as I understand it, is that groups have opinions, but not
always.  Sometimes groups are undecided.  Not every election result is a
group decision, but a good election result should be consistent with
whatever group decisions exist.  By declaring a group to be undecided
when there is no Condorcet Winner, the strength of the consistency
criterion is reduced. 

Similarly, an IRV supporter might declare that there is only a definite
group opinion when there is a first-place majority winner.  Various
methods could be defended by appropriately weak definitions of when a
group decision is made.  The main problem with these definitions,
including the two above, is that they do not always give enough
information to determine a winner (even in non-ties). 

In general, you can think of the group as expressing opinions on certain
pairwise comparisons, and remaining undecided on others.  For example,
you might declare that if X is the CW, the group has decided that it is
superior to each other candidate.  You could do the same if X is the
majority winner. 

But as I say, there are probably many ways you could translate a set of
ballots into a set of pairwise group-decisions.  So, I've come up with
these properties to narrow the search. 

1.  Unanimity.  If every single voter orders their ballot
X1>X2>X3...>Xn, then for any i and j, Xi must be group-chosen over Xj,
iff i>j 

I hope unanimity will be uncontroversial.  This just gets rid of some
very silly possibilities. 

2.  1-Result.  Exactly one ranking should be consistent with the groups
pairwise decisions (except for ties breakable by a single vote). 
Obviously, we don't want groups to be deciding contradictory things, so
at least one ranking must be possible based on their decisions.  And, as
I stated above, my goal is a set of decisions where only one ranking is
possible.  Or you might loosen this to allow multiple possible rankings,
if they give the same winner.  It doesn't make any difference for the
procedures I considered. 

3.  Consistency.  Group decisions should have the property that if we
divide the ballots into two subsets, such that each ballot is in exactly
one subset, and the procedure declares that the each subset has a group
decision of X>Y, then the procedure should find a group-decision of X>Y
in the ballots as a whole.  This just declares that pairwise group
decisions must be consistent, in much the same kind of way that the
consistency criterion does for winners.

Let me show an example of how this could work.  Consider the following
elections 

40: A>B>C 
35: B>C>A 
25: C>A>B 

Most Condorcet methods pick A, but seeing A as a solid group choice
(group-decided over each of B and C) leads to consistency problems
according to property 3.  So, instead I will say only that the group has
declared {A>B,B>C}.  The group has not decided on A vs. C.  However, in
order to give a result consistent with the group-decisions, the election
method must give the ranking A>B>C. 

Now, when I add ballots 

51: A>C>B 
49: C>A>B 

Let's say that I declare the group decisions to be {A>C,A>B,C>B}, using
a rule yet to be revealed.  The election result must be A>C>B. 

Now, let's say the combined ballots give group decisions of
{C>A,C>B,A>B}.  Final ranking C>A>B.  Although the election method still
gives a different result for the combined group than each of the
component groups, pairwise choices given as group decisions are
consistent.  The only group decision common between the subgroups was
A>B, and this was in the final result as well. 

OK, so here's my rule that fits the above properties. A group-decision
of X>Y is made iff 

X pairwise defeats Y with some margin of victory (m). 
And, there is no way to trace a path from Y to X, by moving along
pairwise victories of margin >=m. 

So, for example if we have pairwise victories: A->B [40] B->C [30]
C->A[20], C>A is not a group decision, because here m=20, and I can
travel to B via A->B [40], 40>=20 and then back to C via B->C [30],
30>=20.  But A>B and B>C are both group decisions because no such path
exists. 

I contend that this rule meets the three properties I give above, and in
fact, I will prove it in a later post.  It is the only rule I have found
that meets all three properties, but anyone can try their hand at
finding another.  Note that my use of margins is not just habit;
property 3 is not obeyed by the  winning-votes version of my rule. 

As property 2 says, except for ties, there is only one ranking
consistent with the rule.  Not surprisingly, for my rule, this is the
Ranked Pairs ranking.  But, I have no proof that other rules couldn't be
found in favour of other methods. 

--- 
Blake Cretney 
http://condorcet.org


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